Triangular matrix ring

In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.

If ${\displaystyle T}$ and ${\displaystyle U}$ are rings and ${\displaystyle M}$ is a ${\displaystyle \left(U,T\right)}$-bimodule, then the triangular matrix ring ${\displaystyle R:=\left[{\begin{array}{cc}T&0\\M&U\\\end{array}}\right]}$ consists of 2-by-2 matrices of the form ${\displaystyle \left[{\begin{array}{cc}t&0\\m&u\\\end{array}}\right]}$, where ${\displaystyle t\in T,m\in M,}$ and ${\displaystyle u\in U,}$ with ordinary matrix addition and matrix multiplication as its operations.